A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
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Question about the Poisson process

A service center consists of two servers, each working at an exponential rate of two services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity of ...
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8 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...
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Expectation of a powered complex circular gaussian process

Assuming a complex circular zero-mean gaussian random process (or vector) $\textbf{x}$ $\left(\textbf{x}\sim \mathcal{CN}\left(0,\sigma^2\right)\right)$. $\mathbb{E}\{\textbf{x}\}=0$. The question ...
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Function of Nakagami Distribution

Does anyone know what the distribution of the sum of squared Nakagami is? $$\sum_i^n X_i^2$$ $$X_i\sim \text{Complex Nakagami-m }$$ Is the distribution Erlang? Is the distribution the same as ...
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What is the magnitude of Complex random variable Gaussian Case?

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$ $$X_2 \sim \mathcal{CN}(0,\sigma)$$ If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...
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22 views

Predictable process with stopping time

I would be very gratefull if someone could help me with my question below. Intuitivly I can see that it is correct but I am unsure of how to prove it. Let T be a stopping time in $\mathcal{F}_t$ for ...
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10 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
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26 views

Convergence in $L^2(\Omega\times (0,T))$

Let $$f_i=\exp(\int_0^T h_i(s)\,{\rm d}W_s-1/2\int_0^T h^2_i(s)\,{\rm d}s)$$ where $W_s$ is a brownian motion in a probability space $(\Omega,F,P) $ and $h_i\in L^2(0,T) $. Suppose $F_n\to F$ in ...
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21 views

Stochastic processes on group-valued variables

I have had this question in my head for a long time, and if I don't find out the answer I may explode. So I'm familiar with a basic Ito process, let's say: $dX_t = \mu d t + \sigma d Z_t$. There ...
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25 views

Kolmogorov zero-one law in continuous time?

Let $(X_t : t \geq 0)$ be a stochastic process. Is it necessarily the case that $$P (\limsup_{t \geq 0} X_t \leq a) \in \{ 0,1\}$$ as it is in discrete time? If some conditions are needed on the ...
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19 views

Verifying solution of difference equation?

I have the following difference equation - $2h_{x+1} - 5h + 2h_{x-1} = 0$ for $x = 1, 2, ...., 19$ The boundary conditions are $h_0 = 1$ and $h_{20} = 0$ How would I go about verifying that $h_x = ...
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22 views

Functional representation of adapted jointly measurable stochastic processes

Let $X_t : \Omega \to E, \ t \geq 0$ be continuous-time stochastic process with (Polish) state space $E$ and canonical filtration $\mathcal{F}_t := \sigma(X_u \ | \ u \leq t)$. Let $Y_t : \Omega \to ...
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1answer
15 views

How Do I Find The Permanent of a Double Stochastic Matrix n * n size

I am reading a book on Stochastic Models, and I don't understand this practice questions: A doubly stochastic n × n matrix S has all entries equal to 1/n. The permament of a n × n matrx A is ...
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1answer
52 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
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1answer
12 views

Stochastic Differential equation, expectation and variance

The process is given by $$dU_t=-\gamma U_t\mathrm{d}t+\sigma\mathrm{d}X_t$$ where $U_0 = u$ and $\gamma, \sigma$ are constants. Can you help me out to solve the equation for $U_t$ and find the ...
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1answer
41 views

Integral with respect to brownian motion

Let $f$ be a continuous function on $[0,\infty)$ and $B_t$ be a standard Brownian motion. Define $X_t=\int_0^t f(s) dB(s).$ a) Show that $X_t$ is Gaussian and computer its covariance $C(X_s, X_t)$ ...
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26 views

Measure Preserving tranformation of the space of brownian paths

Let $O$ be an orthogonal transformation of $L2_{[0,\infty)}$. Let $1_{[0,1]}$ be the indicator function for $0 \leq s \leq t$. Also let $B(t)$ be a standard brownian motion. Define $W(t) = ...
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36 views

A question on independence of increments

How could I prove the following? Let $X=(X_t)_{t \in[0,1]}$ be a real-valued stochastic process on a probability space $(\Omega,F,P)$ with $X_0=0$ a.s Show that the following statement as are ...
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1answer
34 views

Battery lifetime as normal distribution?

I want to model battery lifetime, which decrements continuously at every epoch (i.e., work-cycle) in the following way. So it takes values such as 100, 99.7, 99.3, 99.2, ... 0 (a continuous random ...
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72 views

“Back to square one” problem

There's a problem I've been stuck on in preparation for junior programming contest I'm going to participate in. It is as follows: The "back to square one" problem is played on a board that has ...
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29 views

stochastic model [on hold]

I am trying to simulate a model using ssa for a selection of initial values and be able to discuss the results. can anyone let me know if my code make sense? Basically, it's a simple SIR model over ...
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22 views

Equivalent Stopping Times for Brownian Motions

For standard Brownian motion $B$, define stopping time $T_1:=\inf\{t>0: B_t = 3\}$ and $T_2:=\inf\{t>0: B_t = -3\}$ and $T_3 := \min\{T_1, T_2\}$. Can I say that $T_3 = \inf\{t>0, B_t \in ...
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1answer
21 views

How do we compute the mean time spent in transient states of a Markov Chain?

Let $X=\{X_n\}$ be a finite state Markov Chain with the state space $S = \{0,1,2,...,N\}$ such that all the states are transient. The following is the transition matrix. $$ P = \left[\begin{matrix} ...
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10 views

Testing numerical solvers with analytic solution to Ornstein-Uhlenbeck SDE?

I have an SDE I want to solve numerically that is fairly close to the Ornstein-Uhlenbeck process: $$ dx_t=θ(μ−x_t)dt+σdW_t $$ which has analytic solution $$ ...
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1answer
23 views

Integrated Brownian motion: independent stationary increments?

Let $B_t$, $t\in [0,T]$ be a $d$-dimensional standard Brownian motion. Let $\sigma:[0,T] \rightarrow \mathbb R^{d\times d}$ be a deterministic function such that $$\sigma(u) = diag( \sigma_1(u), \dots ...
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1answer
72 views

Laplace transform stopping time

Consider a stochastic differential equation: $$\frac{dX}{dt} = b + \sigma \frac{dW}{dt}, X(0) = x$$ where $b,\sigma$ are constant, $x \in [0,1]$, and $W$ is a Wiener process. Let $\tau = \inf \{ t ...
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Sub-Martingale and Martingale

An integrable sub-martingale $S_t$ with $\mathbb E(S_t)$ being a constant is a martingale. Is this statement true, please? I think so.
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38 views

Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
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1answer
16 views

Brownian Motion and Progressive Process

Let $B_t$ be a Brownian motion. Define sign function as follows. $sign(0) = 0$ and $sign(x) = \frac{x}{|x|}, \forall x \neq 0$. I do not know how to show the following two questions, especially on the ...
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1answer
19 views

Stopping Time and Brownian Motion [closed]

Let $B_t$ be a Brownian motion. Let $a < 0 < b$. Consider $\tau: = \min\{T_a, T_b\}$ where $T_a := \inf\{s \geq 0: B_s \leq a\}$ and $T_b := \inf\{s \geq 0: B_s \geq b\}$, namely, the first ...
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0answers
9 views

Simulating a continous time, inhomogenous Markov chain

What algorithms are used to simulate a time-continous, inhomogenous Markov chain? For the homogenous case, I've found (among others) this reference, which contains a few exact and approximative ...
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1answer
14 views

Piecewise homogeneous Poisson process

Is there a name for a Poisson process which is piecewise homogeneous? I.e. time-homogeneous but with a parameter change each increment. Any references appreciated.
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31 views

Stochastic Processes, requirement at ''source" probability space, is it always an product over $T$?

Let $(\Omega, \mathcal F, P)$ be a probability space, and let $(S, \mathcal S)$ a set $S$ together with a $\sigma$-Algebra over $S$, also let $T$ be some index set, then for each $t \in T$ let $X_t : ...
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11 views

Determining Bounds of a Generating Function of a Stopping Time [duplicate]

Consider the diffusion process $$DX_t=b(X_t)dt+\sigma(X_t)dW_t$$ where $\sigma\sigma*$ is positive definite and $b, \sigma$ are smooth and bounded. Given a one-dimensional domain bounded from 1 side ...
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1answer
26 views

Stochastic integral a Martingale? [closed]

Let $(B_t)$ be a Brownian Motion wrt. to a filtration $(\mathcal{F}_t)$. Set $X_t = \int_0^t B_s d B_s^7.$ Is $X_t$ a Martingale?
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1answer
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Progressively Measurable for Rigth Continuous Adapted Processes

Any adapted and right continuous process $X_t$ is progressively measurable. For the above statement, I found proof in several books. They all have similar argument as follows. For a given $t > ...
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Mean and Covariance of a stochastic process [closed]

x(t)=e^(-t)sin(θ) θ is a random variable uniformly distributed on [0,π] Find a)Mean b)Covariance c)P(t,τ)
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30 views

Integral of a standard brownian motion

I am working on the following problem which is on an introductory chapter of Brownian motion: Let B(t) be the standard Brownian motion. Define $X(t)=[1/\sqrt{t}]\int_{0 \to t}{f(B(s))}ds$ where $f$ ...
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Existence and uniqueness of SDEs depending on the expected value?

I was thinking of general mean-field SDEs. But let us just look at something really simple: $$dX_t = dt + dB_t, \quad X_0=x$$ the solution to this SDE exists in a strong sense and is: $X_t = x + t ...
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How does a Nakagami Random Variable behave?

A Nakagami random variable has the following pdf $$f_{\Omega,m}= \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$ I have two questions regarding this random variable, 1- Is a sum of ...
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Limit of sum of stochastic increments [closed]

Let $(X_t)$ be a continuous local martingale with respect to a filtration $\mathcal{F}_t$. Show that the limit $ \lim_{n\to \infty} \sum_{i=1}^{2^n} (X_{i/2^n}^4 - X_{(i-1)/2^n}^4)^2$ exist in ...
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2answers
28 views

Brownian and Brackets

A continuous martingale with deterministic bracket must be a Brownian motion. Is this statement ture or not, please? If true, how to show it? If not, what is a counter example?
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26 views

How can we prove a relation holds almost surely?

Let's we have a convex function $f:R \rightarrow R$, and g and x are random processes How can we prove the following relation holds almost surely? $\forall z\: \: f(z)\geq ...
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12 views

video lectures on stochastic geometryand point process in wireless communication domain

Can you pls share any link from where i can have video lectures on stochastic geometry with the application in wireless communication. while searching on net i have found some lecture notes in pdf ...
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Why is the canonical filtration of a Brownian motion left-continuous?

Let $\{W_t, t\geq 0\}$ be a Brownian motion, and has a.s. continuous sample paths. Let $\{\mathcal{F}^W_t, t\geq 0\}$ be the canonical filtration, i.e. $\mathcal{F}^W_t=\sigma(W_s, 0\leq s\leq t)$. ...
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21 views

What is the sum-capacity for a non-symmetric interference channel for information theorists?

This question is dedicated for people who are experts in information theory. An interesting result for a two user interference channel in information theory, is the sum-capacity to within one bit. It ...
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Polish spaces and stochastic processes

my professor told me today that stochastic processes taking values in polish spaces are very important, even more important than to study stochastic processes in full generality. Therefore we started ...
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Dose “optional stopping theorem” imply “optional sampling theorem”?

Suppose $X$ is a martingale,$\tau$ and $\sigma$ are two stopping times which satisfy (a)$\sigma\le\tau$ and (b)the "optional stopping theorem" holds,that is to say: $$\mathbb E[X_\sigma]=\mathbb ...
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108 views
+50

How to generate points uniformly distributed on the surface of an ellipsoid?

I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid. If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables ...